Algebraic Dynamics and Transcendental Numbers

Michel Waldschmidt^{1}

Institut de Mathématiques de Jussieu, Université P. et M. Curie, Théorie des Nombres, Case 247, 4, Place Jussieu, F-75252 PARIS Cedex 05 France

Abstract. A first example of a connection between transcendental numbers and complex dynamics is the following. Let p and q be polynomials with complex coefficients of the same degree. A classical result of Böttcher states that p and q are locally conjugates in a neighborhood of ∞: there exists a function f, conformal in a neighborhood of infinity, such that f(p(z)) = q(f(z)). Under suitable assumptions, f is a transcendental function which takes transcendental values at algebraic points. A consequence is that the conformal map (Douady-Hubbard) from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. The underlying transcendence method deals with the values of solutions of certain functional equations.

A quite different interplay between diophantine approximation and algebraic dynamics arises from the interpretation of the height of algebraic numbers in terms of the entropy of algebraic dynamical systems.

Finally we say a few words on the work of J.H. Silverman on diophantine geometry and canonical heights including arithmetic properties of the Hénon map.

1 Transcendental Values of Böttcher Functions

For any complex number c ∈ C, define the polynomial p_{c} ∈ C[z] by p_{c}(z) = z^{2} + c. For n ≥ 1, let p_{c}^{n} be the n-th iterate of p_{c}:

p_{c}^{1}(z) = p_{c}(z) = z^{2} + c, p_{c}^{2}(z) = p_{c}(z^{2} + c) = (z^{2} + c)^{2} + c,
p_{c}^{n}(z) = p_{c}^{n−1}(z^{2} + c) (n ≥ 2).

The Mandelbrot set M can be defined as

M = {c ∈ C | p_{c}^{n}(0) does not tend to ∞ as n → ∞}.

In 1982, A. Douady and J. Hubbard have shown that M is connected. They constructed a conformal map

Φ : C \ M ⟶ {z ∈ C ; |z| > 1}

from the complement of M onto the exterior of the unit disk, which is defined as follows.

For each c ∈ C, there is a unique power series φ_{c} with coefficients in Q(c),

φ_{c}(z) = z + c_{0} + \frac{c_{1}}{z} + \frac{c_{2}}{z^{2}} + … ∈ Q(c)((1/z)),

such that

φ_{c}(z^{2} + c) = φ_{c}(z)^{2}.

For c ∉ M, φ_{c} defines an analytic function near c. Then the above mentioned map Φ is defined by Φ(c) = φ_{c}(c).

According to P.G. Becker, W. Bergweiler and K. Nishioka [2], [3], [11], for any algebraic α ∈ C \ M, the number Φ(α) is transcendental. .

The function φ_{c} is the unique Böttcher function with respect to p_{c} = z^{2}+c and z^{2}. More generally, let

p = az^{d} + ⋯ and q = bz^{d} + ⋯

be two polynomials in C[z] of degree d ≥ 2 and let λ ∈ C satisfy λ^{d−1} = a/b. There exists a unique function f, which is defined and meromorphic in a neighborhood of ∞, such that

lim_{z→∞} \frac{f(z)}{λz} = 1 and f(p(z)) = q(f(z))

for sufficiently large |z|. Such a conjugating function f is called a Böttcher function with respect to p and q.

Assume p and q have algebraic coefficients and are not linearly conjugate to monomials or Chebychev polynomials. Then f is a transcendental function, which takes transcendental values at algebraic points.

This result holds more generally for classes of analytic functions which satisfy certain functional equations. There are two methods to study the transcendence of values of such functions.

The first one originates in the solution, by Th. Schneider, of Hilbert’s seventh problem on the value of the exponential function, which satisfies the functional equation f(z_{1} + z_{2}) = f(z_{1})f(z_{2}). This method can be used to consider other functional equations, like

f(z^{d}) = af(z)^{d} + bz^{h}.

The second method has been introduced by K. Mahler (see [11]) and enables one to prove transcendence results for the values of analytic functions f which are solutions of more general functional equations, like

P (z, f(z), f(z^{d})) = 0.

In the present situation, both methods provide the desired result.

2 Lehmer’s Problem and the Entropy of Algebraic Dynamical Systems

Let F ∈ Z[X] be a monic polynomial of degree d ≥ 1 with complex roots α_{1}, …, α_{d}. Define, for any positive integer n,

Δ_{n}(F) = ∏_{i=1}^{d}(α_{i}^{n} − 1) ∈ Z.

In case F = X − 2 we have Δ_{n}(F) = 2^{n} − 1, and the prime values of the sequence 2^{n} − 1 are the so-called Mersenne primes. In 1933 [7], D.H. Lehmer suggested that the sequence Δ_{n}(F) is likely to produce prime numbers, provided that it grows slowly. If no |α_{i}| is 1, then

lim_{n→∞} \frac{Δ_{n+1}(F)}{Δ_{n}(F)} = ∏_{{1≤i≤n \atop |α_{i}|>1}} |α_{i}|.

More generally, for a polynomial

F = a_{0}X^{d} + ⋯ + a_{d} = a_{0} ∏_{i=1}^{d}(X − α_{i}) ∈ C[X],

define, with K. Mahler,

M(F) = |a_{0}| ∏_{{1≤i≤n \atop |α_{i}|>1}} |α_{i}| = exp ∫_{0}^{1} log |F (e^{2iπt})|dt.

For any polynomial F ∈ C[X], we have M(F) ≥ 1. When F ∈ Z[X], we have M(F) = 1 if and only all its roots α_{i} are either zero or roots of unity. For his calculations, Lehmer used the polynomial F (X) = X^{3} − X − 1. It turns out that this actually is the polynomial having smaller measure > 1 among the non reciprocal polynomials (its root > 1 is the smallest Pisot-Vijiyaraghavan number). For reciprocal polynomials F, that is for F satisfying F (X^{d}) = X^{d}F (1/X), Lehmer said he could not find a polynomial having smaller measure than M(F_{0}) = 1.1762808183 …, with

F_{0}(X) = X^{10} + X^{9} − X^{7} − X^{6} − X^{5} − X^{4} − X^{3} + X + 1 = X^{5}Q(X + (1/X))

and

Q(T) = (T + 1)^{2}(T − 1)(T + 2)(T − 2) − 1.

He asked whether for each c > 1 there is a polynomial F ∈ Z[X] for which 1 < M(F) ≤ c, and this open question is known as Lehmer’s problem.

The number M(F) has a dynamical interpretation, which is a bridge between the notion of height of a polynomial and ergodic theory [6], [12], [5].

For our purposes, an algebraic dynamical system is a continuous endomorphism T : X → X of a metrizable compact topological group. The easiest case, which will be sufficient for our purpose, is the torus X = T^{d} = R^{d}/Z^{d}. Each continuous endomorphism of T^{d} is given by a d × d matrix A_{T} with integer coefficients, and

T x ≡ A_{T} x (mod 1).

Automorphisms are given by a matrix A_{T} with determinant 1 of −1.

An endomorphism T is “ergodic” if, whenever a measurable subset B of X (for a Haar measure μ) satisfies T^{−1}B = B, we have μ(B) = 0 or 1. In the torus case T^{d}, this condition amounts to say that for every square integrable function f, the condition f(T x) = f(x) almost everywhere implies that f is constant almost everywhere.

It follows that an endomorphism T is ergodic is and only if no eigenvalue of A_{T} is a root of unity. So we shall be interested in polynomials (namely the characteristic polynomial χ(A_{T}) of A_{T}) with no root a root of unity.

The set of periodic points of period n under T is

Per_{n}(T) = {x ∈ T^{d} | T^{n}(x) = x}.

If T is ergodic, then the number of periodic points of period n is

|Per_{n}(T)| = | det(A_{T}^{n} − I)| = |Δ_{n}(χ(A_{T}))|.

The topological entropy of T can be defined in terms of the metric: for ϵ > 0 denote by B_{ϵ} the ball around the origin with radius ϵ. Then

h(T) = lim_{ϵ→0} lim_{n→∞} −\frac{1}{n} log μ (∩_{j=0}^{n−1}T^{−j}(B_{ϵ})) .

Denote by λ_{1}, …, λ_{d} the eigenvalues of A_{T} (counting multiplicities). Then Yuzvinskii’s formula reads

h(T) = ∑_{i=1}^{d} log max{1, |λ_{i}]}.

Hence the entropy of T is nothing else than the logarithm of Mahler’s measure of the characteristic polynomial χ(A_{T}) of A_{T}.

Since any monic polynomial X^{d} + a_{1}X^{d−1} + ⋯ + a_{d} is the characteristic polynomial of a matrix, namely

A = ( \begin{matrix}0 \\ ⋮ & & I_{d−1} \\ 0 \\ −a_{d} & −a_{d−1} & ⋯ & −a_{1} \end{matrix} ) ,

it follows that Lehmer’s problem is equivalent to asking: which values in [0, ∞] can occur as an entropy?

According to D.A. Lind, a positive answer to Lehmer’s problem (i.e. the existence of polynomials in Z[X] with arbitrarily small M(F) > 1) is equivalent to the existence of a continuous endomorphism of the infinite torus T^{Z} with finite entropy.

Interesting problems occur when one tries to replace the torus R/Z by an elliptic curve [1].

This section has been prepared with the help of Paola D’Ambros.

3 Canonical Heights and Dynamical Systems

Define the absolute multiplicative height of a polynomial F ∈ Z[X] of degree d > 0 by

H(F) = M(F)^{1/d}

and the absolute multiplicative height of an algebraic number α by

H(α) = H(F)

where F ∈ Z[X] is the minimal polynomial of α over Z. The name is motivated by the property

H(α^{n}) = H(α)^{n}

for any algebraic number α and any positive integer n. Hence this height H behaves nicely with respect to the polynomials ϕ(X) = X^{n}.

J.H. Silverman [18] introduced a height function which behaves nicely for an arbitrary rational function ϕ with algebraic coefficients, viewed as a map P_{1}(\overline{Q}) → P_{1}(\overline{Q}).

Definition. Let ϕ ∈ \overline{Q}(X) be a rational function of degree n ≥ 2. The ϕcanonical height of an algebraic number is

Ĥ_{ϕ}(α) = lim_{r→∞}(ϕ^{r}(α))^{1/n^{r}},  (∗)

where ϕ^{r} = ϕ ∘ ϕ^{r−1} and ϕ^{0} is the identity.

This construction has been introduced by Tate in his work on Abelian varieties, and has been extended to this general context by Silverman in a series of papers. He proved:

The limit (∗) defining Ĥ_{ϕ}(α) exists, and

Ĥ_{ϕ}(ϕα) = Ĥ_{ϕ}(α)^{n}.

Moreover

Ĥ_{ϕ}(α) ≥ 1

for any α ∈ \overline{Q}, with equality if and only if α is pre-periodic for ϕ.

Recall that α is a pre-periodic point for ϕ if the orbit {α, ϕα, ϕ^{2}α, …} contains only finitely many points.

The natural generalization of Lehmer’s conjecture to this more general setting had been raised by P. Moussa, J-S. Geronimo and D. Bessis in 1984 [10].

The rational map

ϕ(X) = \frac{(X^{2} − 1)^{2}}{4X^{3} + 4X}

corresponds to the duplication map on the elliptic curve Y^{2} = X^{3} + X. For this map, and more generally for the rational maps corresponding to multiplication by an integer on an elliptic curve, partial results towards this Lehmertype conjecture are known (M. Laurent, D.W. Masser and S.W. Zhang, M. Hindry and J. Silverman).

Variants of this construction have been proposed, mainly by J.H. Silverman. In [13], he defined heights on K3 surfaces using two involutions which generate an infinite group of automorphisms. With G.S. Call in [4], he did the same on general varieties V by using a morphism ϕ : V → V and a divisor D for which ϕ^{∗}D is linearly equivalent to αD with α > 1. In [14], he considered a variety V related with the Hénon map

ϕ : A^{2} → A^{2}, ϕ(X, Y) = (Y, Y^{2} + aX + b)

as follows: blowing up each of the points (1 : 0 : 0) and (0 : 1 : 0) three times, one obtains a variety V so that both ϕ and ϕ^{−1} extend to morphisms V → P^{2}. For P ∈ A^{2}(\overline{Q}), the relation

h(ϕ^{n}P) + h(ϕ^{−n}P) ≥ (2^{n} + 2^{−n})(h(P) − c) + 2c

holds with some constant c = c(ϕ). Silverman used this inequality to prove that ϕ has only finitely many periodic points with rational coordinates and to count the growth rate of points in an infinite orbit.

The author wishes to thank Michel Planat for the excellent organization of this conference.

References

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